How to prevent ambiguity that arises from multiple conditional statements in one sentence?

Question

Conditional statements are important when writing mathematical proofs/articles. However, I have found that it is sometimes important to be able to have multiple conditional statements in the same sentence when trying to prove something. Consider statements $a$, $b$, $c$ and $d$ and consider the following example sentence that is dependent on the result of some previous sentence (this is why the sentence starts with "Hence"):

• "Hence, $a$ if $b$ and $c$ if $d$."

This can be interpreted as meaning ($a$ if $b$) and ($c$ if $d$), (which is usually my intention), or as (($b$ and $c$) if $d$) and ($a$ if ($b$ and $c$)). Another seemingly valid interpretation is that ($a$ if ($b$ and $c$)), which is only true if $d$.

How might one restructure the sentence so that the intended meaning is clear (($a$ if $b$) and ($c$ if $d$))? One could separate the conditional statements into their own sentences. But the sentence starts with "Hence" and it wouldn't make sense to separate out the conditional statements into separate sentences that each begin with "Hence". E.g. consider the modified version of the original sentence:

• "Hence, $a$ if $b$. Hence, $c$ if $d$."

This version of the sentence does not make sense because the second sentence in this version is not dependent on the result of the first sentence in this version, it is dependent on the sentence before the original sentence.

I have also considered the use of semi-colons, colons or additional commas. However, it is not completely clear to me if these methods make sense. Consider the following sentence:

• "Hence, $a$ if $b$, and $c$ if $d$."

It seems like this extra comma removes some ambiguity. As does the following sentence:

• "Hence: $a$ if $b$; $c$ if $d$."

However, I'm unsure if this is a correct use of semi-colons and colons, and I'm unsure if the use of the comma completely removes any ambiguity. Could the comma be interpreted as simply an oxford comma in the list of hypotheses for the first conditional statement? I've also seen the suggestion that you can parenthesize the sentences. However, I never see this in any journals and articles in my field.

Any ideas or strategies that can help remove ambiguity?

Answer 1

Why restrict yourself to just one sentence?

For each quadruple $(a,b,c,d) \in A$, both the following two conditions hold. (1) If $b=1$ then $a=1$. (2) If $d=1$ then $c=1$.

(You have asked several questions on this site about writing mathematics. Two useful general principles are (1) words are usually better than formal logical statements, and (2) clear is better than short.)

Note: This is an answer to the original formulation of the question, which the OP has now changed. The principle here remains the same.

Answer 2

• "Hence, for any $(a,b,c,d) \in A$, $a=1$ if $b=1$ and $c=1$ if $d=1$."

Non-ambiguous alternatives:

• Hence, for every $(a,b,c,d) \in A$, it is both the case that $a=1$ if $b=1$ and that $c=1$ if $d=1.$

• Hence, for every $(a,b,c,d) \in A$, it is both the case that $b=1$ implies $a=1$ and that $d=1$ implies $c=1.$

• Ethan's suggestion in the other answer.

This can be interpreted as $(a=1 \text{ if } b=1)$ and $(c=1 \text{ if } d=1)$ (which is usually my intention),

or as $((b=1 \land c=1) \text{ if } d=1)$ together with $(a=1 \text{ if } (b=1 \land c=1)).$

Another seemingly valid interpretation is: $(a=1 \text{ if } (b=1 \land c=1))$ is true if $d=1.$

While the sentence $$P⟹Q⟹R$$ would be informally understood as "$(P⟹Q)$ and $(Q⟹R)$", the sentence $$P \text{ if } Q \text{ if } R$$ is simply ambiguous, and not typically understood as "$(P$ if $Q)$ and $(Q$ if $R)$".

The most charitable reading of $$a=1 \text{ if } b=1 \text{ and } c=1 \text{ if } d=1$$ is your intended one, as opposed to (a=1 if (b=1 and c=1)) if d=1, ((a=1 if b=1) and c=1) if d=1, etc.